Edexel_2017_5

Edexel_2017_5

(a)

Let \theta be the acute angle between \Pi_1 and \Pi_2, then:

\begin{align*}\cos \theta &= \frac {\begin{pmatrix}1\\-2\\-3\end{pmatrix} \cdot \begin{pmatrix}6\\1\\-4\end{pmatrix}} {\left|\begin{pmatrix}1\\-2\\-3\end{pmatrix}\right| \left|\begin{pmatrix}6\\1\\-4\end{pmatrix}\right|} \\ & = \frac {6 - 2 + 12} {\sqrt {1^2 + 2^2 + 3^2} \sqrt {6^2 + 1 + 4^2}} \\ & = \frac {16} {\sqrt {742}}\end{align*}

So:

\displaystyle \theta = \arccos \left(\frac {16} {\sqrt {742}}\right) = 54.0288\ldots^\circ = 54^\circ \, (\text {nearest integer})

(b)

A line perpendicular to \Pi_1 is parallel to the normal \begin{pmatrix}1\\-2\\-3\end{pmatrix}, since it passes through \begin{pmatrix}2 \\ 3 \\-1\end{pmatrix} it has vector equation:

\mathbf r = \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}2 \\ 3 \\-1\end{pmatrix} + \lambda \begin{pmatrix}1\\-2\\-3\end{pmatrix}

Substituting this into the equation for \Pi_2 to find the intersection we have:

6(2 + \lambda) + (3 - 2 \lambda) - 4(-1 - 3\lambda) = 12 + 6 \lambda + 3 - 2 \lambda + 4 + 12 \lambda = 16\lambda + 19 = 7

So:

\displaystyle \lambda = -\frac {12} {16} = -\frac 3 4

Giving:

\displaystyle Q = \left(2 - \frac 3 4, 3 + 2 \times \frac 3 4, -1 + 3 \times \frac 3 4\right) = \left(\frac 5 4, \frac 9 2, \frac 5 4\right)

©

If the plane \Pi_3 is perpendicular to \Pi_1 and \Pi_2 it is parallel to \begin{pmatrix}1 \\ -2 \\ -3\end{pmatrix} and \begin{pmatrix}6 \\ 1 \\ -4\end{pmatrix}, meaning that it has normal \begin{pmatrix}1 \\ -2 \\ -3\end{pmatrix} \times \begin{pmatrix}6 \\ 1 \\ -4\end{pmatrix} = \begin{pmatrix}11 \\ -14 \\ 13\end{pmatrix}. Since the point \left(\frac 5 4, \frac 9 2, \frac 5 4\right) lies on the plane, the plane has equation:

\displaystyle \mathbf r \cdot \begin{pmatrix}\frac 5 4\\ \frac 92 \\ \frac 5 4\end{pmatrix} = \begin{pmatrix}11 \\ -14 \\ 13\end{pmatrix} \cdot \begin{pmatrix}\frac 5 4\\ \frac 92 \\ \frac 5 4\end{pmatrix} = \frac {55} 4 - \frac {126} 2 + \frac {65} 4 = -33

ie.

\mathbf n = \begin{pmatrix}\frac 5 4 \\ \frac 9 2 \\ \frac 5 4\end{pmatrix}

and:

p = -33

multiples are fine.

Absolute LaTeX tekkers!!!